# Langevin dynamcs MCMC

Randomly exploring the posterior space.

## Annealed

Jolicoeur-Martineau et al. (2022); Song and Ermon (2020a); Song and Ermon (2020b)

Generative Modeling by Estimating Gradients of the Data Distribution | Yang Song

## Incoming

See log concave distributions for a family of distributions where this works especially well. Rob Salomone explains this well; see Hodgkinson, Salomone, and Roosta (2019). Holden Lee, Andrej Risteski introduce the connection between log-concavity and convex optimisation.

$x_{t+\eta} = x_t - \eta \nabla f(x_t) + \sqrt{2\eta}\xi_t,\quad \xi_t\sim N(0,I).$

Left-field, Max Raginsky, Sampling Using Diffusion Processes, from Langevin to Schrödinger:

the Langevin process gives only approximate samples from $$\mu$$. I would like to discuss an alternative approach that uses diffusion processes to obtain exact samples in finite time. This approach is based on ideas that appeared in two papers from the 1930s by Erwin Schrödinger in the context of physics, and is now referred to as the Schrödinger bridge problem.

## References

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Garbuno-Inigo, Alfredo, Franca Hoffmann, Wuchen Li, and Andrew M. Stuart. 2020. SIAM Journal on Applied Dynamical Systems 19 (1): 412–41.
Ge, Rong, Holden Lee, and Andrej Risteski. 2020. arXiv:1812.00793 [Cs, Math, Stat], September.
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Jolicoeur-Martineau, Alexia, Rémi Piché-Taillefer, Ioannis Mitliagkas, and Remi Tachet des Combes. 2022. In.
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Song, Yang, and Stefano Ermon. 2020a. In Advances In Neural Information Processing Systems. arXiv.
———. 2020b. In Advances In Neural Information Processing Systems. arXiv.
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